Correct length-biased sampling from family-size survey

Imagine you're in a town and stop by a school, where you ask 100 children: “How many kids are in your family?” The responses break down like this:

• 50 children report a family size of 1
• 20 children report a family size of 2
• 30 children report a family size of 3 Next, you pick a random house in that town, knock on the door, and ask: “How many children reside here?” For this scenario, assume each family has at least one child and that family sizes can only be 1, 2, or 3. Question: What is your best guess for the probability that a randomly selected house contains exactly 1 child? You should state any assumptions you need regarding how the 100 children were selected (for instance, uniformly at random from all children in the town). Overview: This problem tests your grasp of length-biased sampling and drawing inferences when sampling bias is present—specifically, how sampling children as individuals leads to different family-size probability estimates than sampling households.